Hidden Zero

Author

Eric W. Weisstein
May 12, 2020

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/HiddenZero.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HiddenZero.html.

©2020 Wolfram Research, Inc. except for portions noted otherwise

Without Simplify

(((1-e))^2)-(e^2)+2 e-1

((1-E)^2+2E-1-E^2)

Simplify[%]

(((1-π))^2)-(π^2)+2 π-1

((1-Pi)^2+2Pi-1-Pi^2)

Simplify[%]

1-(20/(Sqrt[(50-10 (Sqrt[5])) (10+2 (Sqrt[5]))]))

(1-(20/(Sqrt[(50-10 (Sqrt[5])) (10+2 (Sqrt[5]))])))

Simplify[%]

ϕ

GoldenRatio==(1+(Sqrt[5]))/2

With Parameters

2 (tanh^(-1))((Sqrt[x]))+log((2/((Sqrt[x])+1))-1)

FullSimplify[2 ArcTanh[Sqrt[x]] + Log[-1 + 2/(1 + Sqrt[x])]]

-((dn(u|m))^2)-m ((sn(u|m))^2)+1

FullSimplify[expr=1 - JacobiDN[u, m]^2 - m JacobiSN[u, m]^2]

Γ(n+1,x)-((e^(-x)) (x^n)+n Γ(n,x))

Gamma[n+1, x]-(n Gamma[n, x] + x^n Exp[-x])

log((1+(1/z)) (Sqrt[(((1/z)-1)/(1+(1/z)))])+(1/z))-log((Sqrt[(1/z)-1]) (Sqrt[1+(1/z)])+(1/z))

FullSimplify[ Log[Sqrt[(-1 + z^(-1))/(1 + z^(-1))]*(1 + z^(-1)) + z^(-1)] - Log[Sqrt[-1 + z^(-1)]*Sqrt[1 + z^(-1)] + z^(-1)]]

(((((log(a))/(W(log(a))))))^((log(a))/(W(log(a)))))-a

FullSimplify[-a + (Log[a]/ProductLog[Log[a]])^(Log[a]/ProductLog[Log[a]])]

((2 K((((Sqrt[1-z])-1)/(2 (Sqrt[1-z])))))/(π ((1-z)^(1/4))))-((2 K(((2 (Sqrt[z]))/((Sqrt[z])+1))))/(π (Sqrt[(Sqrt[z])+1])))

FullSimplify[expr=(2*EllipticK[(-1+Sqrt[1-z])/(2*Sqrt[1-z])])/(Pi*(1-z)^(1/4))-(2*EllipticK[(2*Sqrt[z])/(1+Sqrt[z])])/(Pi*Sqrt[1+Sqrt[z]])]

FullSimplify[FunctionExpand[expr]]

y ((((y/p)))^(1/(p-1)))-p ((((y/p)))^(p/(p-1)))

FullSimplify[expr=y(y/p)^(1/(-1 + p)) - p(y/p)^(p/(-1 + p))]

FullSimplify[FunctionExpand[expr]]

(Subscript[, 2])(Subscript[F, 1])(a,b;c;1)-((Subscript[(c-b), -a])/(Subscript[(c), -a]))

FullSimplify[expr=Hypergeometric2F1[a,b,c,1]-Pochhammer[c-b,-a]/Pochhammer[c,-a],
  Re[c-a-b]>0]

θ(b+a x)-θ(sgn(a) ((b/a)+x))

FullSimplify[expr=UnitStep[a x + b] - UnitStep[Sign[a](x + b/a)]]

(Sqrt[((a^2)/((a^2)-1))])-(1/(Sqrt[1-(1/(a^2))]))

FullSimplify[-(1/Sqrt[1 - a^(-2)]) + Sqrt[a^2/(-1 + a^2)],Im[a] != 0]

(i^n) (Subscript[U, n])(-(i/2))-(Subscript[F, n+1])

FullSimplify[expr=I^n*ChebyshevU[n,-I/2]-Fibonacci[n+1],n∈Integers]

FullSimplify[FunctionExpand[expr],n∈Integers]

-2 a E(1-((b^2)/(a^2)))+a E(1-((b^2)/(a^2)))+b E(1-((a^2)/(b^2)))

FullSimplify[a EllipticE[1 - b^2/a^2] + b EllipticE[1 - a^2/b^2]-2a EllipticE[1-b^2/a^2],a>0&&b>0]

(Sqrt[(((a-b))^3) z])-(a-b) (Sqrt[(a-b) z])

FullSimplify[expr=-(a - b) Sqrt[(a - b) z] + Sqrt[(a - b)^3 z], a - b > 0]

cos(2 A)-cos(2 (B+C))

FullSimplify[Cos[2*A] - Cos[2*(B + C)], A+B+C == Pi]

(2 (Sqrt[-9 (x^6)+36 (x^2)+16 (Sqrt[3])]) (27 (x^4)-18 (Sqrt[3]) (x^2)+3 (3^(1/4)) (3 x+2 (3^(3/4))) (Sqrt[2 (3^(3/4))-3 x]) (Sqrt[2 (3^(1/4))-(Sqrt[3]) x]) (x^2)+2 (3^(3/4)) (3 x+2 (3^(3/4))) (Sqrt[2 (3^(3/4))-3 x]) (Sqrt[2 (3^(1/4))-(Sqrt[3]) x])-72))/(3 (((-3 (Sqrt[3]) (x^4)+6 (x^2)+8 (Sqrt[3])))^2))

FullSimplify[expr=(2*Sqrt[16*Sqrt[3]+36*x^2-9*x^6]*(-72-18*Sqrt[3]*x^2+27*x^4+2*3^(3/4)*Sqrt[2*3^(3/4)-3*x]*(2*3^(3/4)+3*x)*Sqrt[2*3^(1/4)-Sqrt[3]*x]+3*3^(1/4)*Sqrt[2*3^(3/4)-3*x]*x^2*(2*3^(3/4)+3*x)*Sqrt[2*3^(1/4)-Sqrt[3]*x]))/(3*(8*Sqrt[3]+6*x^2-3*Sqrt[3]*x^4)^2)]

FullSimplify[expr,x>0]

(2^(a/4)) a Γ(-(a/4)) Γ((a/8)) sin(((a π)/8))+4 Γ((1/2)-(a/8)) (Sqrt[π])

FullSimplify[expr=4*Sqrt[Pi]*Gamma[1/2-a/8]+2^(a/4)*a*Gamma[-a/4]*Gamma[a/8]*Sin[(a*Pi)/8]]

(1/(Sqrt[(x^2)+1]))((Sqrt[(x^2)+1]) (Sqrt[2 x (x-(Sqrt[(x^2)+1]))+1])+x ((Sqrt[2 x (x-(Sqrt[(x^2)+1]))+1])-(Sqrt[((x+(Sqrt[(x^2)+1]))/x)]) (Sqrt[((Sqrt[(x^2)+1])/x)-1])))

FullSimplify[expr=(Sqrt[1 + x^2]*Sqrt[1 + 2*x*(x - Sqrt[1 + x^2])] + x*(-(Sqrt[(x + Sqrt[1 + x^2])/x]*Sqrt[-1 + Sqrt[1 + x^2]/x]) + Sqrt[1 + 2*x*(x - Sqrt[1 + x^2])]))/Sqrt[1 + x^2]]

No Parameters

24808: (Subscript[, 2])(Subscript[F, 1])((1/4),(1/4);1;(1/64))-(Sqrt[(2/(7 π))]) (Sqrt[((Γ((1/7)) Γ((2/7)) Γ((4/7)))/(Γ((3/7)) Γ((5/7)) Γ((6/7))))])

FullSimplify[expr=Hypergeometric2F1[1/4, 1/4, 1, 1/64] -
Sqrt[2/(7Pi)] Sqrt[(Gamma[1/7] Gamma[2/7] Gamma[4/7])/(Gamma[3/7]Gamma[5/7] Gamma[6/7])]]

FullSimplify[FunctionExpand[expr]]

N[expr,10^3]

Developer`ZeroQ[expr]

25366:(((Sqrt[π]) F((cos^(-1))(((1-(Sqrt[3]))/((Sqrt[3])+1)))|(1/4) ((Sqrt[3])+2)))/(2 (3^(1/4)) Γ((1/3)) Γ((7/6))))-1

FullSimplify[expr=(Sqrt[Pi]*EllipticF[ArcCos[(1-Sqrt[3])/(1+Sqrt[3])],(2+Sqrt[3])/4])/(2*3^(1/4)*Gamma[1/3]*Gamma[7/6])-1]

FullSimplify[TrigToExp[FunctionExpand[expr]]]

N[expr,10^3]

Developer`ZeroQ[expr]

23664:(2/3) (4 (cot^(-1))((Sqrt[2]))+(cot^(-1))(2 (Sqrt[2]))+2 (cot^(-1))(2 (Sqrt[2])-(Sqrt[3]))+2 (cot^(-1))(2 (Sqrt[2])+(Sqrt[3])))-π

FullSimplify[expr=(2 (4 ArcCot[Sqrt[2]] + ArcCot[2 Sqrt[2]] + 2 ArcCot[2 Sqrt[2] - Sqrt[3]] + 2 ArcCot[2 Sqrt[2] + Sqrt[3]]))/3-Pi]

FullSimplify[TrigToExp[expr]]

N[expr,10^4]

Developer`ZeroQ[expr]

28164:(((6/(3-2 (5^(1/4))))-1)^(1/4))-coth(((log(5))/8))

FullSimplify[expr=(-1+6/(3-2*5^(1/4)))^(1/4)-Coth[Log[5]/8]]

FullSimplify[TrigToExp[expr]]

N[%,10^3]

Developer`ZeroQ[expr]

29449: (Sqrt[(((Sqrt[3])+1)/(4 (Sqrt[2])))+(1/2)])-cos((π/24))

FullSimplify[expr=Sqrt[1/2+(1+Sqrt[3])/(4 Sqrt[2])]-Cos[Pi/24]]

FullSimplify[TrigToExp[%]]

Developer`ZeroQ[expr]

34043:2 ((csc^(-1))((((1-(e^(-i)))^(1/4))/(Sqrt[(1/2) ((Sqrt[1-(e^(-i))])-1)])))+(csc^(-1))((((1-(e^i))^(1/4))/(Sqrt[(1/2) ((Sqrt[1-(e^i)])-1)]))))

FullSimplify[expr=2*(ArcCsc[(1 - E^(-I))^(1/4)/Sqrt[1/2*(-1 + Sqrt[1 - E^(-I)])]] + ArcCsc[(1 - E^I)^(1/4)/Sqrt[1/2*(-1 + Sqrt[1 - E^I])]])-π/2]

FullSimplify[TrigToExp[expr]]

FullSimplify[2 (ArcCsc[((((1-(E^(-I))))^(1/4))/(Sqrt[(1/2) (-1+(Sqrt[1-(E^(-I))]))]))]+ArcCsc[((((1-(E^I)))^(1/4))/(Sqrt[(1/2) (-1+(Sqrt[1-(E^I)]))]))])]

FullSimplify[TrigToExp[%]]

N[expr,10^3]

Developer`ZeroQ[expr]

50298:2 (cot^(-1))(3 (Sqrt[3]))+(cot^(-1))((5/(Sqrt[3])))+(cot^(-1))((7/(Sqrt[3])))+(cot^(-1))((19/(Sqrt[3])))-(π/3)

FullSimplify[expr=ArcCot[5/Sqrt[3]] + ArcCot[7/Sqrt[3]] +
ArcCot[19/Sqrt[3]] + 2*ArcCot[3*Sqrt[3]]-π/3]

FullSimplify[TrigToExp[expr]]

N[expr,10^3]

Developer`ZeroQ[expr]

57875: (cos^(-1))((17/81))-8 (tan^(-1))((Sqrt[2]))+2 π

expr=FullSimplify[2Pi+ArcCos[17/81]-8ArcTan[Sqrt[2]]]

FullSimplify[TrigToExp[%]]

Developer`ZeroQ[expr]

N[expr,10^4]

58004: 8 i π log((Sqrt[2])+1)+π log((((2 (2 (Sqrt[2])+3)))^(-6 i)))+π log((((((2-(Sqrt[2])))^(9 i)) ((((Sqrt[2])+2))^(7 i)))/(((10-7 (Sqrt[2])))^(3 i))))-i π log(7 (Sqrt[2])+10)

FullSimplify[expr=8*I*Pi*Log[1 + Sqrt[2]] + Pi*Log[((2 - Sqrt[2])^(9*I)*(2 + Sqrt[2])^(7*I))/(10 - 7*Sqrt[2])^(3*I)] + Pi*Log[(2*(3 + 2*Sqrt[2]))^(-6*I)] - I*Pi*Log[10 + 7*Sqrt[2]]]

FullSimplify[TrigToExp[expr]]

N[expr,10^3]

Developer`ZeroQ[expr]

58505: (1/8) (π (i (-8 (sinh^(-1))(1)+log(128)+2)-6 π)+log(((((10-7 (Sqrt[2])))^((3 i) π))/(((2-(Sqrt[2])))^((17 i) π)))))+((3 (π^2))/4)

FullSimplify[expr=(Pi*(-6*Pi+I*(2-8*ArcSinh[1]+Log[128]))+Log[(10-7*Sqrt[2])^((3*I)*Pi)/(2-Sqrt[2])^((17*I)*Pi)])/8+3π^2/4]

FullSimplify[TrigToExp[expr]]

FullSimplify[ComplexExpand[expr]]

N[expr,10^3]

Developer`ZeroQ[expr]

66510: 3 (csch^(-1))(2 (Sqrt[2+(Sqrt[5])]))-(1/2) (sinh^(-1))(2)

FullSimplify[3ArcCsch[2(Sqrt[2+(Sqrt[5])])]-ArcSinh[2]/2]

FullSimplify[TrigToExp[%]]

66602:(1/16) (log(((2321^(3 (-6 i+(Sqrt[2])))) (((-1560 i (Sqrt[2])-721))^(Root[((#1)^4)-28 ((#1)^2)+484&,1])) (((1560 i (Sqrt[2])-721))^(9 i-3 (Sqrt[2]))) (((11 i+5 (Sqrt[3])))^(64 i)) (((-100007080 i (Sqrt[2])+60693952 i (Sqrt[3])+53866252 (Sqrt[6])-177513785))^(11 i)) (((((-30560 i (11 (Sqrt[2])+4 (Sqrt[3]))+47506 (Sqrt[6])+261283)/1533667)))^(22 i)))/((5^(6 (Sqrt[2]))) (9^(Sqrt[-384 i (Sqrt[2])-4078])) (((-2 i (Sqrt[2])-1))^(152 i)) (((2 (-7 i+4 (Sqrt[2])) (23 i+55 (Sqrt[3]))))^(32 i))))+6 log(3) (Sqrt[2])+3 log(58025) (Sqrt[2])+108 π)

FullSimplify[expr=(1/16)*(108*Pi+6*Sqrt[2]*Log[3]+3*Sqrt[2]*Log[58025]+Log[(2321^(3*(-6*I+Sqrt[2]))*(-721-1560*I*Sqrt[2])^Root[484-28*#1^2+#1^4&,1]*(-721+1560*I*Sqrt[2])^(9*I-3*Sqrt[2])*(11*I+5*Sqrt[3])^(64*I)*(-177513785-100007080*I*Sqrt[2]+60693952*I*Sqrt[3]+53866252*Sqrt[6])^(11*I)*((261283+47506*Sqrt[6]-30560*I*(11*Sqrt[2]+4*Sqrt[3]))/1533667)^(22*I))/(5^(6*Sqrt[2])*9^Sqrt[-4078-384*I*Sqrt[2]]*(-1-2*I*Sqrt[2])^(152*I)*(2*(-7*I+4*Sqrt[2])*(23*I+55*Sqrt[3]))^(32*I))])]//Timing

FullSimplify[TrigToExp[expr]]

N[expr,10^3]

Developer`ZeroQ[expr]

Releaux Tetrahedron Volume

expr=(3/(22 (Sqrt[2])))+((2 (Sqrt[2]))/11)+((8 π)/3)+(11/2) ArcTan[(1/(4 (Sqrt[2])))]-(11/2) ArcTan[(3/(Sqrt[2]))]-(11 I (2 I-3 (Sqrt[2])) ArcTan[((-6240-56517 I (Sqrt[2])-5808 I (Sqrt[-7+6 I (Sqrt[2])]))/(45081 I+1200 (Sqrt[2])))])/(4 (Sqrt[2 (-7+6 I (Sqrt[2]))]))+(11 I (2 I-3 (Sqrt[2])) ArcTan[(387072-1446912 I (Sqrt[2])-278784 I (Sqrt[6 (-7+6 I (Sqrt[2]))]))/(1378560 I+225792 (Sqrt[2]))])/(4 (Sqrt[2 (-7+6 I (Sqrt[2]))]))+(11 (-2 I-3 (Sqrt[2])) ArcTan[((2 (792448-37632 I (Sqrt[2])))/(129024 I-482304 (Sqrt[2])-61952 (Sqrt[3 (7+6 I (Sqrt[2]))])))])/(4 (Sqrt[2 (7+6 I (Sqrt[2]))]))-((2 I (-I+(Sqrt[3])) ArcTan[((-12-19 I (Sqrt[3])-8 I (Sqrt[-1+I (Sqrt[3])]))/(31 I+4 (Sqrt[3])))])/(Sqrt[(1/2) (-1+I (Sqrt[3]))]))+((2 I (-I+(Sqrt[3])) ArcTan[((288-336 I (Sqrt[3])-288 I (Sqrt[6 (-1+I (Sqrt[3]))]))/(528 I+96 (Sqrt[3])))])/(Sqrt[(1/2) (-1+I (Sqrt[3]))]))-((2 (I+(Sqrt[3])) ArcTan[((-336+96 I (Sqrt[3])-96 (Sqrt[6 (1+I (Sqrt[3]))]))/(-96 I+400 (Sqrt[3])))])/(Sqrt[(1/2) (1+I (Sqrt[3]))]))-(11 I (-2 I-3 (Sqrt[2])) ArcTanh[((2 (21103 I+200 (Sqrt[2])))/(2080 I+18839 (Sqrt[2])+5082 (Sqrt[2 (7+6 I (Sqrt[2]))])))])/(4 (Sqrt[2 (7+6 I (Sqrt[2]))]))+((2 I (I+(Sqrt[3])) ArcTanh[((19 I-4 (Sqrt[3])+16 I (Sqrt[1+I (Sqrt[3])]))/(-4 I+7 (Sqrt[3])))])/(Sqrt[(1/2) (1+I (Sqrt[3]))]))+(((-I+(Sqrt[3])) Log[9])/(Sqrt[(1/2) (-1+I (Sqrt[3]))]))+((I (I+(Sqrt[3])) Log[9])/(Sqrt[(1/2) (1+I (Sqrt[3]))]))-(((-I+(Sqrt[3])) Log[144])/(Sqrt[(1/2) (-1+I (Sqrt[3]))]))-((I (I+(Sqrt[3])) Log[144])/(Sqrt[(1/2) (1+I (Sqrt[3]))]))+((11 (2 I-3 (Sqrt[2])) Log[1089])/(8 (Sqrt[2 (-7+6 I (Sqrt[2]))])))+((11 I (-2 I-3 (Sqrt[2])) Log[1089])/(8 (Sqrt[2 (7+6 I (Sqrt[2]))])))-((11 (2 I-3 (Sqrt[2])) Log[30976])/(8 (Sqrt[2 (-7+6 I (Sqrt[2]))])))-((11 I (-2 I-3 (Sqrt[2])) Log[30976])/(8 (Sqrt[2 (7+6 I (Sqrt[2]))])))-(11 (2 I-3 (Sqrt[2])) (Log[3]+Log[-139 I+16 (Sqrt[2])-28 I (Sqrt[2 (-7+6 I (Sqrt[2]))])]))/(8 (Sqrt[2 (-7+6 I (Sqrt[2]))]))+(11 (2 I-3 (Sqrt[2])) (Log[256]+Log[-59 I+6 (Sqrt[2])-4 I (Sqrt[3 (-7+6 I (Sqrt[2]))])]))/(8 (Sqrt[2 (-7+6 I (Sqrt[2]))]))-(11 I (-2 I-3 (Sqrt[2])) (Log[3]+Log[-529 I+48 (Sqrt[2])-32 I (Sqrt[7+6 I (Sqrt[2])])]))/(8 (Sqrt[2 (7+6 I (Sqrt[2]))]))+(11 I (-2 I-3 (Sqrt[2])) (Log[768]+Log[-73 I+6 (Sqrt[2])-6 I (Sqrt[6 (7+6 I (Sqrt[2]))])]))/(8 (Sqrt[2 (7+6 I (Sqrt[2]))]))+((-I+(Sqrt[3])) (Log[48]+Log[-7 I+2 (Sqrt[3])-3 I (Sqrt[2 (-1+I (Sqrt[3]))])]))/((Sqrt[(1/2) (-1+I (Sqrt[3]))]))-(((-I+(Sqrt[3])) Log[-9 I+4 (Sqrt[3])-8 I (Sqrt[3 (-1+I (Sqrt[3]))])])/(Sqrt[(1/2) (-1+I (Sqrt[3]))]))-((I (I+(Sqrt[3])) Log[-17 I+4 (Sqrt[3])-4 I (Sqrt[1+I (Sqrt[3])])])/(Sqrt[(1/2) (1+I (Sqrt[3]))]))+(I (I+(Sqrt[3])) (Log[48]+Log[-11 I+2 (Sqrt[3])-3 I (Sqrt[6 (1+I (Sqrt[3]))])]))/((Sqrt[(1/2) (1+I (Sqrt[3]))]))-(1/12 (3 Sqrt[2]-49 Pi+162 ArcTan[Sqrt[2]]));

N[expr,10^3]

FullSimplify[TrigToExp[expr]]//Timing

Developer`ZeroQ[expr]

Releaux Volume Again

FullSimplify[expr=((8/3)+((23 I)/8)) π+(1/16) (4 (Sqrt[2])+88 ArcCot[4 (Sqrt[2])]-44 ArcCot[(4931/(Sqrt[89292194+21463200 I (Sqrt[2])]))]-64 ArcCot[(7/(Sqrt[3 (-83+60 I (Sqrt[3]))]))]+64 ArcCot[(49/(Sqrt[1839+300 I (Sqrt[3])]))]+64 ArcCot[(1009/(Sqrt[1090523+137144 (Sqrt[6])+8 I (64149 (Sqrt[2])-17612 (Sqrt[3]))]))]-64 ArcCot[(163/(Sqrt[5995+3664 (Sqrt[6])+16 I (6581 (Sqrt[2])+5332 (Sqrt[3]))]))]-6 I (-3 I+(Sqrt[2])) ArcTan[(39/(20 (Sqrt[2])))]-88 ArcTan[(3/(Sqrt[2]))]+44 ArcTan[(4201/(Sqrt[11004674-4499880 I (Sqrt[2])]))]+44 ArcTan[Root[170116652304+4469014944 ((#1)^2)+46810091160 ((#1)^4)+2739187944 ((#1)^6)+787195249 ((#1)^8)&,8]]+6 (Sqrt[2]) Log[3]-53 I Log[11]+3 (Sqrt[2]) Log[58025]+Log[(25^(Root[19321+206 ((#1)^2)+((#1)^4)&,1])) (211^(-9 I)) (729^(Root[6889+158 ((#1)^2)+((#1)^4)&,1])) (((-721-1560 I (Sqrt[2])))^(Root[484-28 ((#1)^2)+((#1)^4)&,1])) ((((1/97) (485+312 I (Sqrt[2])+36 I (Sqrt[3])-100 (Sqrt[6]))))^(22 I)) ((((1/973) (1763+1402 I (Sqrt[2])-514 I (Sqrt[3])+1730 (Sqrt[6]))))^(32 I)) (((-177513785-100007080 I (Sqrt[2])+60693952 I (Sqrt[3])+53866252 (Sqrt[6])))^(11 I))])-1/12 (3 Sqrt[2]-49 Pi+162 ArcTan[Sqrt[2]])]

FullSimplify[TrigToExp[expr]]

N[expr,10^3]

expr=((8/3)+((23 I)/8)) π+(1/16) (4 (Sqrt[2])+88 ArcCot[4 (Sqrt[2])]-44 ArcCot[(4931/(Sqrt[89292194+21463200 I (Sqrt[2])]))]-64 ArcCot[(7/(Sqrt[3 (-83+60 I (Sqrt[3]))]))]+64 ArcCot[(49/(Sqrt[1839+300 I (Sqrt[3])]))]+64 ArcCot[(1009/(Sqrt[1090523+137144 (Sqrt[6])+8 I (64149 (Sqrt[2])-17612 (Sqrt[3]))]))]-64 ArcCot[(163/(Sqrt[5995+3664 (Sqrt[6])+16 I (6581 (Sqrt[2])+5332 (Sqrt[3]))]))]-6 I (-3 I+(Sqrt[2])) ArcTan[(39/(20 (Sqrt[2])))]-88 ArcTan[(3/(Sqrt[2]))]+44 ArcTan[(4201/(Sqrt[11004674-4499880 I (Sqrt[2])]))]+44 ArcTan[Root[170116652304+4469014944 ((#1)^2)+46810091160 ((#1)^4)+2739187944 ((#1)^6)+787195249 ((#1)^8)&,8]]+6 (Sqrt[2]) Log[3]-53 I Log[11]+3 (Sqrt[2]) Log[58025]+Log[(25^(Root[19321+206 ((#1)^2)+((#1)^4)&,1])) (211^(-9 I)) (729^(Root[6889+158 ((#1)^2)+((#1)^4)&,1])) (((-721-1560 I (Sqrt[2])))^(Root[484-28 ((#1)^2)+((#1)^4)&,1])) ((((1/97) (485+312 I (Sqrt[2])+36 I (Sqrt[3])-100 (Sqrt[6]))))^(22 I)) ((((1/973) (1763+1402 I (Sqrt[2])-514 I (Sqrt[3])+1730 (Sqrt[6]))))^(32 I)) (((-177513785-100007080 I (Sqrt[2])+60693952 I (Sqrt[3])+53866252 (Sqrt[6])))^(11 I))])-1/12 (3 Sqrt[2]-49 Pi+162 ArcTan[Sqrt[2]]);

Developer`ZeroQ[expr]

((((1-(Sqrt[5])))^128)/340282366920938463463374607431768211456)+(((((Sqrt[5])+1))^128)/340282366920938463463374607431768211456)-562882766124611619513723647

FullSimplify[ expr=(1 - Sqrt[5])^128/
  340282366920938463463374607431768211456 + 
 (1 + Sqrt[5])^128/340282366920938463463374607431768211456-562882766124611619513723647]

Developer`ZeroQ[expr]

Developer`SystemOptions["SimplificationOptions" ]

Developer`SetSystemOptions["SimplificationOptions" ->{"SimplifyMaxExponents" -> 1000}];

FullSimplify[expr]

Expand[%]

Developer`SetSystemOptions["SimplificationOptions" ->{"SimplifyMaxExponents" -> 100}];

((((1-(Sqrt[5])))^128)/340282366920938463463374607431768211456)+((251728825683549488150424261 (Sqrt[5]))/2)-(562882766124611619513723647/2)

FullSimplify[expr=-(562882766124611619513723647/2)+(251728825683549488150424261*Sqrt[5])/2+(1-Sqrt[5])^128/340282366920938463463374607431768211456]

Expand[%]

Developer`ZeroQ[expr]

Developer`SetSystemOptions["SimplificationOptions" ->{"SimplifyMaxExponents" -> 1000}];

FullSimplify[expr=-(562882766124611619513723647/2)+(251728825683549488150424261*Sqrt[5])/2+(1-Sqrt[5])^128/340282366920938463463374607431768211456]

Developer`SetSystemOptions["SimplificationOptions" ->{"SimplifyMaxExponents" -> 100}];

Implemented

(sin^(-1))((4/5))+(tan^(-1))(7)-((3 π)/4)

FullSimplify[expr=ArcSin[4/5] + ArcTan[7] - 3*Pi/4]

(cos^(-1))((1/(Sqrt[26])))-(tan^(-1))(5)

FullSimplify[expr=ArcCos[1/Sqrt[26]]-ArcTan[5]]

(i^(((-i))^i))-(i^(i^(-i)))

FullSimplify[expr = I^(-I)^I - I^I^(-I)]

(cot^(-1))((5/(Sqrt[3])))+(cot^(-1))(3 (Sqrt[3]))-(π/6)

FullSimplify[expr=ArcCot[5/Sqrt[3]]+ArcCot[3*Sqrt[3]]-Pi/6]

(cot^(-1))((Sqrt[2]))-(1/2) (π-2 (tan^(-1))((Sqrt[2])))

FullSimplify[expr=Subtract @@ {ArcCot[Sqrt[2]], (1/2)*(Pi - 2*ArcTan[Sqrt[2]])}]

Numercize with small imaginary part

-((-1)^(1/7))+(((-1))^(6/7))

-(((-1))^(1/7))+(((-1))^(6/7))//N

1+Sqrt[(Sin[Pi/16]-ToRadicals[Sin[Pi/16]])]

expr=1+Sqrt[(Sin[Pi/16]-ToRadicals[Sin[Pi/16]])]

Table[N[expr,prec],{prec,{MachinePrecision,10}}]

Root[-59 + 110*#1 - 60*#1^2 + 10*#1^3 &, 1, 0] // ToRadicals

expr=ToRadicals[(NumericalApproximation[Root, (RowDefault["0.953", FEExpressionsNumericalApproximationElider]), Root[-59+110 #1-60 ((#1)^2)+10 ((#1)^3)&,1], ])]

Table[N[expr,prec],{prec,{MachinePrecision,10}}]

1+Sqrt[(Pi+E)^2-(Pi^2+2 E Pi+E^2)]

expr=1+Sqrt[(Pi+E)^2-(Pi^2+2 E Pi+E^2)]

Table[N[expr,prec],{prec,{MachinePrecision,10}}]